A novel construction of lackadaisical quantum walk

Abstract This paper constructs a new model for the lackadaisical quantum walk and analyzes its features. Traditionally, the particle state is represented by a three-dimensional complex vector, with the Grover operator as the evolution unitary. Analytical methods calculate position probability distributions, but computational complexity and error grow exponentially with steps. To address this, the paper introduces an evolution unitary operator composed of three operators for left movement, right movement, and staying. The staying state corresponds to multiple self-loops at position vertices. A path analysis method is employed to calculate position probabilities and characteristics in polynomial time. Lackadaisical quantum walks differ from classical walks, increasing self-loops enhances diffusion, following a Gaussian mixture model. Numerical experiments reveal lackadaisical characteristics such as slower diffusion, localization, and ballistic diffusion.